06.11.2007 (Tuesday)

Asymptotics of biorthogonal polynomials associated to the two-matrix model

Regular Seminar Maurice Duits (KU Leuven)

16:00 Brunel U.
room M128

I will report on some joint work with Arno Kujlaars on the two-matrix model in random matrix theory. In this model one is interested in the eigenvalue statistics for two NxN hermitian Matrices, M1 and M2, taken random from exp(-NTrV(M1)-NTrW(M2)+tauTrM1M2)dM1dM2. Here V and W are two polynomials of even degree and tau is called coupling contant. The eigenvalues statistics of M1 and M2 can be expressed in terms of certain biorthogonal polynomials. Therefore, if one can find asymptoticss for the biorthogonal polynomials then one also knowns the asymptotics for the eigenvalue statistics. Asymptotic results for the biorthogonal polynomials are known in the physics literature, however, they are without rigurous proofs. I will discuss a rigorous approach for a special case. The key element is a Deift-Zhou steepest descent analysis for a 4x4 Riemann-Hilbert problem.